\newproblem{lay:5_5_23}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 5.5.23}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $A$ be an $n\times n$ real matrix with the property that $A^T=A$, let $\mathbf{x}$ be any vector in $\mathbb{C}^n$, and let $q=(\mathbf{x}^*)^TA\mathbf{x}$. Show that
	$q$ is a real number.
}{
  % Solution
	We need to show that $q^*=q$.
	\begin{center}
		$\begin{array}{rcll}
			q^*&=&((\mathbf{x}^*)^TA\mathbf{x})^*&[(AB)^*=A^*B^*; (\mathbf{x}^*)^T=(\mathbf{x}^T)^*)]\\
			   &=&\mathbf{x}^TA^*\mathbf{x}^*& [\text{by hypothesis A is real}] \\
			   &=&\mathbf{x}^TA\mathbf{x}^*& [q^T=q; (ABC)^T=C^TB^TA^T] \\
				 &=&(\mathbf{x}^*)^TA^T\mathbf{x} & [\text{by hypothesis } A^T=A] \\
				 &=&(\mathbf{x}^*)^TA\mathbf{x} & \\
				 &=&q
		\end{array}$
	\end{center}
}
\useproblem{lay:5_5_23}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
